The Hypercircle in Mathematical Physics.

We develop the nonconforming virtual element method for linear plate bending problems. A class of nonconforming virtual elements is constructed, which is C0-continuous. Like the classical nonconforming plate elements, it relaxes the continuity requirement for the function space to some extent. Further, the virtual element is constructed for any order of accuracy and adapts to complicate element geometries. We present a general framework on the error analysis for the nonconforming virtual element method, highlighting the main difference with the conforming one.

The nonconforming virtual element method for plate bending problems

This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree capable of providing a consistent approximation of Sobolev spaces Hm in Rn (with n ≥ m ≥ 1) and also a convergent (nonconforming) finite element space for 2m-th-order elliptic boundary value problems in Rn. For this class of finite element spaces, the geometric shape is n-simplex, the shape function space consists of all polynomials with a degree not greater than m, and the degrees of freedom are given in terms of the integral averages of the normal derivatives of order m − k on all subsimplexes with the dimension n − k for 1 ≤ k ≤ m. This sequence of spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases. The finite element spaces constructed in this paper constitute the only class of finite element spaces, whether conforming or nonconforming, that are known and proven to be convergent for the approximation of any 2m-th-order elliptic problems in any Rn, such that n ≥ m ≥ 1. Finite element spaces in this class recover the nonconforming linear elements for Poisson equations (m = 1) and the well-known Morley element for biharmonic equations (m = 2).

/pdf/minimal-finite-element-spaces-for-2m-th-order-partial-3pm8bhu8vj.pdf

Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$

A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance

In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form ( − Δ ) p u = f , p ≥ 1 . More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems, and prove an abstract result that states the convergence of the method in suitable norms.

The conforming virtual element method for polyharmonic problems

In this paper, a new class of Zienkiewicz-type non-conforming finite element, in n spatial dimensions with n ≥ 2, is proposed. The new finite element is proved to be convergent for the biharmonic equation.

http://www.math.pku.edu.cn/teachers/wangm/nm5599r.pdf

A new class of Zienkiewicz-type non-conforming element in any dimensions

We investigated how current least-squares reverse time migration (LSRTM) methods perform on subsalt images. First, we compared the formulation of data-domain vs. image-domain least-squares migration (LSM), as well as methods using single-iteration approximation vs. iterative inversion. Next, we examined the resulting subsalt images of several LSRTM methods applied on both synthetic and field data. Among our tests, we found image-domain single-iteration LSRTM methods, including an extension from Guitton’s (2004) method in the curvelet domain, not only compensated for amplitude loss due to poor illumination caused by complex salt bodies, but also produced subsalt images with fewer migration artifacts in the field data. By contrast, an iterative inversion method showed its potential for broadening bandwidth in the subsalt, but was less effective in reducing noise. Based on our understanding, we summarize the current state of LSRTM for subsalt imaging, especially between singleiteration and iterative LSRTM methods.

Least-squares RTM: Reality and possibilities for subsalt imaging

In this paper, we consider the use of $$H(\mathrm{div })$$H(div) elements in the velocity---pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair $$\mathrm{RT}_0$$RT0---$$\mathrm{P}_0$$P0, which is known to be suboptimal, and render the error estimate optimal by the symmetry of the grids and by the superconvergence result of Lagrange interpolant. By enlarging $$\mathrm{RT}_0$$RT0 such that it becomes a modified $$\mathrm{BDM}$$BDM-type element, we develop a new discretization $$\mathrm{BDM}_1^\mathrm{b}$$BDM1b---$$\mathrm{P}_0$$P0. We, therefore, generalize the classical MAC scheme on rectangular grids to triangular grids and retain all the desirable properties of the MAC scheme: exact divergence-free, solver-friendly, and local conservation of physical quantities. Further, we prove that the proposed discretization $$\mathrm{BDM}_1^\mathrm{b}$$BDM1b---$$\mathrm{P}_0$$P0 achieves the optimal convergence rate for both velocity and pressure on general quasi-uniform grids, and one and half order convergence rate for the vorticity and a recovered pressure. We demonstrate the validity of theories developed here by numerical experiments.

/pdf/convergence-analysis-of-triangular-mac-schemes-for-two-4rqrafj3g1.pdf

Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations

An efficient multigrid solver for the Oseen problems discretized by Marker and Cell (MAC) scheme on staggered grid is developed in this paper. Least squares commutator distributive Gauss-Seidel (LSC-DGS) relaxation is generalized and developed for Oseen problems. Residual overweighting technique is applied to further improve the performance of the solver and a defect correction method is suggested to improve the accuracy of the discretization. Some numerical results are presented to demonstrate the efficiency and robustness of the proposed solver.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/AEFB7EC99AD030A7D6726A7CC37B698D/S1004897915000124a.pdf/div-class-title-a-multigrid-solver-based-on-distributive-smoother-and-residual-overweighting-for-oseen-problems-div.pdf

A Multigrid Solver based on Distributive Smoother and Residual Overweighting for Oseen Problems

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